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<h1 style="text-align: justify">IPredict Time-series Forecasting Methods</h1>
		<p style="text-align: justify">IPredict offers a wide selection of 
		time-series forecasting algorithms, regression forecasting, smoothing and curve or model fitting algorithms. Together 
		with all classical algorithms you will find proprietary and advanced 
		algorithms with superior performance in statistical forecasting.<br>
		Take a look at IPredict <a href="http://www.ipredict.it/Benchmarks.aspx">benchmarks</a> 
		to see the difference in performance of some of these algorithms.<br>
		To see how these methods are applied to everyday financial forecasting 
		take a look at our free online <a href="http://www.ipredict.it/OnlineDemo.aspx">
		stock market prediction</a> or <a href="http://www.ipredict.it/Public/DemoWF.zip">download</a> 
		the example program that applies Wavelet Forecasting to financial data.</p>
		<p style="text-align: justify">All proprietary algorithms (14. Holt 
Winter's Modified Multiple 
		Seasonalities, 26. Haar Denoising, 27. Daubechies Linear Denoising, 
28. Daubechies Exponential Denoising, 29. Wavelet Forecasting, 34. 
Fractal Projection, 35. Active Moving Average)
		are subject to copyright and all rights are reserved to IPredict.</p>
		<p style="text-align: justify">The most important feature of IPredict is 
		the Excel wizard that allows you, without forecasting expertise or statistics 
		knowledge, to produce a reliable forecast in minutes. The wizard 
		automates the time-series forecasting tasks, computes the best parameters given the 
		time-series and allows you to choose visually the best approach for your 
		problem. The best fit is calculated using the most appropriate technique 
		by ranking using a user selected <a href="http://www.ipredict.it/ErrorStatistics.aspx">statistical 
		error</a>.</p>
		<h1 style="text-align: justify">Classical Algorithms</h1>
		<h4 style="text-align: justify">1. Simple Moving Average</h4>
		<p style="text-align: justify">The Simple Moving Average smooth past 
		data by arithmetically averaging over a specified period and projecting 
		forward in time. This is normally considered a smoothing algorithm and 
		has poor forecasting results in most cases.</p>
		<h4 style="text-align: justify">2. Geometric Moving Average</h4>
		<p style="text-align: justify">The Geometric Moving Average smooth past 
		data by geometrically averaging over a specified period and projecting 
		forward in time. This is normally considered a smoothing algorithm and 
		has poor forecasting results in most cases.</p>
		<h4 style="text-align: justify">3. Triangular Moving Average</h4>
		<p style="text-align: justify">The Triangular Moving Average is a 
		weighted moving average with weights that form a triangular shape. The 
		projection technique is the same of the Simple Moving Average. This is 
		normally considered a smoothing algorithm and has poor forecasting 
		results in most cases.</p>
		<h4 style="text-align: justify">4. Parabolic Moving Average</h4>
		<p style="text-align: justify">The Parabolic Moving Average is a 
		weighted moving average with weights that form a parabolic shape. The 
		projection technique is the same of the Simple Moving Average. This is 
		normally considered a smoothing algorithm and has poor forecasting 
		results in most cases.</p>
		<h4 style="text-align: justify">5. Double Moving Average</h4>
		<p style="text-align: justify">The Double Moving Average applies in 
		sequence for two times the Simple Moving Average algorithm. This is 
		normally considered a smoothing algorithm and has poor forecasting 
		results in most cases.</p>
<h4 style="text-align: justify">6. Exponential Moving Average</h4>
		<p style="text-align: justify">The Exponential Moving Average is 
		summarized by the equation:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = αX<sub>t </sub>
		+ (1-α)X'<sub>t-1<br>
		</sub>it is a weighted moving average with weights that decrease 
		exponentially going backwards in time. This is normally considered a 
		smoothing algorithm and has poor forecasting results in most cases.</p>
		<h4 style="text-align: justify">7. Double Exponential Moving Average</h4>
		<p style="text-align: justify">The Double Exponential Moving Average 
		applies the Exponential Moving Average twice. This is normally 
		considered a smoothing algorithm and has poor forecasting results in 
		most cases.<br>
		If the equation of the single exponential moving average can be 
		expressed as in (6):<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = αX<sub>t </sub>
		+ (1-α)X'<sub>t-1<br>
		</sub>then the equation of the double exponential moving average can be 
		expressed as:<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; X''<sub>t</sub> = αX<sup>'</sup><sub>t </sub>
		+ (1-α)X''<sub>t-1</sub></p>
		<h4 style="text-align: justify">8. Holt's Double Exponential</h4>
		<p style="text-align: justify">Holt's Double Exponential is similar to 
		the Double Exponential Moving Average. It allows you to specify the two 
		smoothing constants used in the process. It is useful for data in a 
		simple linear trend.</p>
		<h4 style="text-align: justify">9. Triple Exponential Moving Average</h4>
		<p style="text-align: justify">The Triple Exponential Moving Average 
		applies three times the Exponential Moving Average. This is normally 
		considered a smoothing algorithm and has poor forecasting results in 
		most cases.<br>
		If the equation of the single exponential moving average can be 
		expressed as in (6):<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = αX<sub>t </sub>
		+ (1-α)X'<sub>t-1<br>
		</sub>and the equation of the double exponential moving average can be 
		expressed as in (7):<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; X''<sub>t</sub> = αX<sup>'</sup><sub>t </sub>
		+ (1-α)X''<sub>t-1<br>
		</sub>then the equation of the triple exponential moving average can be 
		expressed as:<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; X'''<sub>t</sub> = αX<sup>''</sup><sub>t </sub>
		+ (1-α)X'''<sub>t-1</sub></p>
		<h4 style="text-align: justify">10. Holt's Triple Exponential</h4>
		<p style="text-align: justify">Holt's Triple Exponential is the 
		classical Holt's forecasting algorithm. It is useful for data in a 
		simple linear trend.</p>
		<h4 style="text-align: justify">11. Adaptive Response Rate Exponential 
		Smoothing</h4>
		<p style="text-align: justify">The Adaptive Exponential Smoothing 
		automatically adjusts the smoothing parameters based on the forecast 
		error. This is normally considered a smoothing algorithm and has poor 
		forecasting results in most cases.</p>
<h4 style="text-align: justify">12. Holt Winter's Additive</h4>
		<p style="text-align: justify">The Holt Winter's Additive method is 
		applicable when the time series contains a seasonal component. This 
		method assumes the time series is composed by a linear trend and a 
		seasonal cycle, it constructs three statistically correlated series 
		(smoothed, seasonal and trend) and projects forward the identified trend 
		and seasonality.</p>
		<h4 style="text-align: justify">13. Holt Winter's Multiplicative</h4>
		<p style="text-align: justify">Like Holt Winter's Additive this method 
		can be applied to a seasonal time series. The model assumes that the 
		components of the time series (smoothed, seasonal and trend) are 
		multiplied together giving as result a more 'active' time series. </p>
		<h4 style="text-align: justify">14. Holt Winter's Modified Multiple Seasonalities</h4>
		<p style="text-align: justify">The Modified Multiple Seasonalities is a 
		proprietary algorithm based on Holt Winter's algorithm that can take 
		into consideration multiple seasonalities. This model is especially 
		suited for financial time series. Two versions of this algorithm are now 
		available with different computational logic.</p>
		<h4 style="text-align: justify">15. Additive Decomposition</h4>
		<p style="text-align: justify">Additive Decomposition computes the 
		decomposition of the time series into its components, trend, 
		seasonality, cyclical and error. It projects the identified parts to the 
		future and sums the resulting projection to form the forecast. The model 
		is assumed to be additive (that is all parts are summed up to give the 
		forecast).<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = T<sub>t </sub>
		+ S<sub>t </sub>
		+ C<sub>t </sub>
		+ ε<sub>t<br>
		</sub>where T is the trend, S the seasonality, C the cycle and
		ε the error.</p>
		<h4 style="text-align: justify">16. Multiplicative Decomposition</h4>
		<p style="text-align: justify">Multiplicative Decomposition like 
		Additive Decomposition computes the decomposition of the time series 
		into its components, trend, seasonality, cyclical and error and then 
		projects to the future. The model is assumed to be multiplicative (that 
		is all parts are multiplied by each other to give the forecast).<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = T<sub>t </sub>* S<sub>t </sub>* C<sub>t </sub>
		* ε<sub>t<br>
		</sub>where T is the trend, S the seasonality, C the cycle and
		ε the error.</p>
		<h4 style="text-align: justify">17. Sparse Series Croston's Exponential</h4>
		<p style="text-align: justify">This is a very useful algorithm for 
		sparse time series. The model is equivalent to an Exponential Moving 
		Average both in quantities and in time.</p>
		<h1 style="text-align: justify">Curve and Bayesian Model Fitting</h1>
		<h4 style="text-align: justify">18. Linear Trend / Regression</h4>
		<p style="text-align: justify">The Linear Trend fits the time series to 
		a straight line and projecting forward in time.<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = m*t + q</p>
		<h4 style="text-align: justify">19. Linear Trend And Additive 
		Seasonality</h4>
		<p style="text-align: justify">The Linear Trend and Additive Seasonality 
		assume the data is made of a linear trend plus a single additive 
		seasonality. It then fits the equation to the data using a Bayesian fit 
		and projects forward in time.<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = m*t + A*sin(ωt) 
		+ q</p>
		<h4 style="text-align: justify">20. Linear Trend And Multiplicative 
		Seasonality</h4>
		<p style="text-align: justify">The Linear Trend and Multiplicative 
		Seasonality assume the data is made of a linear trend plus a single 
		multiplicative seasonality. It then fits the equation to the data using 
		a Bayesian fit and projects forward in time.<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = m*t * A*sin(ωt) 
		* q</p>
		<h4 style="text-align: justify">21. Linear Trend And Multiple 
		Seasonalities</h4>
		<p style="text-align: justify">The Linear Trend and Multiple 
		Seasonalities assume the data is made of a linear trend plus multiple 
		additive or multiplicative seasonalities. It then fits the equation to 
		the data using a Bayesian fit and projects forward in time. It is 
		especially useful for financial time series.</p>
		<h4 style="text-align: justify">22. Polynomial</h4>
		<p style="text-align: justify">The Polynomial algorithm fits a 
		polynomial equation (up to the desired order) to the data using a 
		Bayesian fit and projects forward in time. You must be careful not to 
		over fit the data with a very long polynomial.<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = A + B*t + C*t<sup>2</sup> + D*t<sup>3</sup> 
		+ ...</p>
		<h4 style="text-align: justify">23. Logarithmic</h4>
		<p style="text-align: justify">The Logarithmic algorithm fits a 
		logarithmic equation to the data using a Bayesian fit and projects 
		forward in time.<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = A + B*log(t)</p>
		<h4 style="text-align: justify">24. Exponential</h4>
		<p style="text-align: justify">The Exponential algorithm fits an 
		exponential equation to the data using a Bayesian fit and projects 
		forward in time.<br>
		The model equation is:<br>
&nbsp;&nbsp;&nbsp;&nbsp; X'<sub>t</sub> = A + B*exp(t)</p>
		<h1 style="text-align: justify">Wavelet Smoothing and Forecasting</h1>
		<h4 style="text-align: justify">25. Frequency Identification</h4>
		<p style="text-align: justify">One of the biggest issues when dealing 
		with algorithms that require seasonality indexes is to compute the 
		seasonalities. All competitors require you to compute these magic 
		numbers and even the most advanced packages aren't able to tell this 
		simple and intuitive figure: the seasonality of a time series. The 
		Frequency Identification algorithm computes the frequencies that are 
		inside the input time series.</p>
		<h4 style="text-align: justify">26. Haar Denoising</h4>
		<p style="text-align: justify">Removing noise from a time series is 
		always difficult and current algorithms (averages, exponential averages, 
		etc...) always introduce a lag in data or change the statistical 
		properties of the underlying time series. Haar Denoising is able to 
		remove the noise that is in the time series using the Haar Wavelet 
		transform and a proprietary algorithm. </p>
		<h4 style="text-align: justify">27. Daubechies Linear Denoising</h4>
		<p style="text-align: justify">Like Haar Denoising the Daubechies Linear 
		Denoising applies a Daubechies Wavelet transform and a proprietary 
		algorithm to linearly denoise the time series.</p>
		<h4 style="text-align: justify">28. Daubechies Exponential Denoising</h4>
		<p style="text-align: justify">The denoising in this case decreases 
		exponentially with the wavelet power.</p>
		<h4 style="text-align: justify">29. Wavelet Forecasting</h4>
		<p>This algorithm uses the Daubechies Wavelet transform to 
		produce a forecast with a proprietary algorithm.</p><h1 style="text-align: justify">
		Additional Functions and Algorithms</h1>
		<h4 style="text-align: justify">30. Random Number Generation</h4>
		<p style="text-align: justify">Special care must be taken into account 
		when generating test beds for ideas and models. Usually the stock Rand() 
		Excel function is not enough. IPredict provides four statistically 
		independent Uniform Random Number Generators and one Gaussian Generator.</p>
		<h4 style="text-align: justify">31. Wavelet Transforms</h4>
		<p style="text-align: justify">This is the base of the Wavelet 
		forecasting algorithm. The wavelet transform is the base of a lot of 
		today's algorithms in the field of Digital Signal Processing, Quantum 
		Mechanics, Image Processing and Speech Recognition among others. The 
		forward and inverse Daubechies and Haar transforms are included.</p>
		<h4 style="text-align: justify">32. Fourier Transforms</h4>
		<p style="text-align: justify">
            These are the classical Fourier Transforms for forecasting. IPredict includes Sine,
            Cosine and Fast Fourier
		Transforms.</p>
		<h4 style="text-align: justify">33. Hurst Exponent</h4>
		<p style="text-align: justify">The estimate of the Hurst Exponent is 
		very important to help understand the properties of a time-series and 
		how much it looks like a random walk (i.e. how much the time-series is 
		predictable). Three methods are provided based on Daubechies, Haar and 
		Rescaled Range algorithms.</p>
<h1 style="text-align: justify">Advanced Algorithms</h1>

<h4 style="text-align: justify">34. Fractal Projection</h4>
<p style="text-align: justify">
The Fractal Projection algorithm represents a new approach to forecasting. 
This algorithm is able to project to the future a pattern stretching it in times or values as appropriate.
</p>

<h4 style="text-align: justify">35. Active Moving Average</h4>
<p style="text-align: justify">
The Active Moving Average belongs to the class of Exponential Moving Averages but its calculation
is quadratic in nature and is much quicker than standard Moving
    Averages in "following" the original signal.
</p>


<h1 style="text-align: justify">Kernel Smoothing</h1>

Kernel smoothing weights every single data point in a time-series with weights coming from a generating function.

<h4 style="text-align: justify">36. Gaussian Kernel Smoothing</h4>
<p style="text-align: justify">
The Gaussian Kernel Smoothing is a classical Kernel Smoothing algorithm. 
    <br>
The kernel function is the following:<br>
K(t) = e<sup>-λ * t<sup>2</sup></sup>
</p>

<h4 style="text-align: justify">37. Hilbert Kernel Smoothing</h4>
<p style="text-align: justify">
The Hilbert Kernel Smoothing function is:<br>
K(t) = -π / t
</p>

<h4 style="text-align: justify">38. Triangle Kernel Smoothing</h4>
<p style="text-align: justify">
The Triangle Kernel Smoothing function is:<br>
K(T) = 1 - Abs(t) 
</p>

<h4 style="text-align: justify">39. Epanechnicov Kernel Smoothing</h4>
<p style="text-align: justify">
The Epanechnicov Kernel Smoothing function is:<br>
K(t) = 3/4 * (1 - t<sup>2</sup>)
</p>

<h4 style="text-align: justify">40. Quartic Kernel Smoothing</h4>
<p style="text-align: justify">
The Quartic Kernel Smoothing function is:<br>
K(t) = 15/16 * (1 - t<sup>2</sup>)<sup>2</sup>
</p>

<h4 style="text-align: justify">41. Triweight Kernel Smoothing</h4>
<p style="text-align: justify">
The Triweight Kernel function is:<br>
K(t) = 35/32 * (1 - t<sup>2</sup>)<sup>3</sup>
</p>

<h4 style="text-align: justify">42. Cosine Kernel Smoothing</h4>
<p style="text-align: justify">
The Cosine Kernel function is:<br>
K(t) = -π/4 * Cos(π/2 * t)
</p>


<h4 style="text-align: justify">43. Savitsky-Golay Smoothing</h4>
<p style="text-align: justify">
This algorithm is a smoothing filter that essentially applies a 
polynomial regression of a certain degree to a time-series. 
The advantage of the Savitsky-Golay filter is that it tends to preserve 
certain features of the time-series like local minima and maxima.
</p>
          

<h4>44. Spline Smoothing</h4>
<p>
Spline smoothing consists in computing the spline that approximates the 
input data and projecting it to internal or external points. It is 
useful when compressing or expanding data before a projection is done.
</p>


<br>





    

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                </td>
                <td>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Benchmarks.aspx">Holt Winter's, Series Decomposition and Wavelet Benchmarks</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/UseOfMAInForecasting.aspx">Use of the Moving Average in Time-series Forecasting</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Denoising.aspx">Denoising Techniques</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Performance.aspx">Computational Performance</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/MovingAverage.aspx">Moving Averages</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/ActiveMovingAverage.aspx">Active Moving Average</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/FractalProjection.aspx">Fractal Projection</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/Regression.aspx">Multiple Regression</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/PrincipalComponentAnalysis.aspx">Principal Component Analysis</a>.<br>
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/OptionsBlackScholesMerton.aspx">Options Pricing with Black-Scholes</a>.<br>                       
                    <img style="vertical-align: middle; border: 0pt none; height: 5px; width: 5px; margin-left: 5px;" src="ForecastingMethods.aspx-Dateien/arrow-sm.jpg" alt="">
                    <a href="http://www.ipredict.it/Methods/Preprocessing.aspx">Time-series preprocessing</a>.<br>                </td>
            </tr>
        </tbody>
    </table>
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